Current algebras and higher genus CFT partition functions Roberto Volpato Institute for Theoretical Physics – ETH Zurich ZURICH, RTN Network 2009 Based on: M. Gaberdiel and R.V., arXiv: 0903.4107 [hep-th] M. Gaberdiel, C. Keller, R.V., work in progress
2-D CFT 2-D Conformal Field Theory on a surface of genus Amplitudes depend on the choice of a complex structure
Partition functions Moduli space Partition function on Riemann surface corresponds to Ex:
Motivations How much information in the partition function? Genus 1 spectrum of the theory Genus 2,3,… ? Can we reconstruct a CFT from partition functions? [Friedan, Schenker ’87] Which functions on are CFT partition functions? Modular invariance, factorisation, … what else? Applications to Ads/CFT correspondence 3-d pure quantum gravity, chiral gravity, ... [Witten ’07] [Li, Song, Strominger ’08]
Main results The affine Lie algebra of currents in a CFT is uniquely determined by its PFs (and representations are strongly constrained) M. Gaberdiel and R.V., JHEP 0906:048 (2009) [arXiv:0903.4107] Constraints for meromorphic unitary theories from genus 2 partition functions M. Gaberdiel, C. Keller and R.V., work in progress
How can we obtain information on a CFT from its partition function? Factorisation properties under degeneration limits
Degeneration limit
Factorization
Multiple degenerations…
…2n-point amplitudes n parameters 2n-point correlators
Can we obtain directly all correlators? NO Can we reconstruct the whole CFT? Open problem [Friedan, Schenker ’87] Can we reconstruct the algebra of currents? [M. Gaberdiel and R.V. ’09] YES
Kac-Moody affine algebras Currents (conformal weight 1) Mode expansion (sphere) Kac-Moody affine algebra Level Structure constants
Assumptions We only consider unitary bosonic meromorphic self-dual CFTs (but results hold more generally) Lattice theories: CFT of free chiral bosons on even unimodular lattice Example: 16 chiral bosons in heterotic strings ( and )
General procedure Consider a genus g partition function Take the degeneration limit to a torus
General procedure Consider a genus g partition function Take the degeneration limit to a torus Consider the term in the power expansion of ( )
General procedure Consider a genus g partition function Take the degeneration limit to a torus Consider the term in the power expansion of ( ) Integrate the coefficient over non-trivial cycle
General procedure Consider a genus g partition function Take the degeneration limit to a torus Consider the term in the power expansion of ( ) Integrate the coefficient over non-trivial cycle Expand in powers of Space of conf. weight h
General procedure Consider a genus g partition function Take the degeneration limit to a torus Consider the term in the power expansion of ( ) Integrate the coefficient over non-trivial cycle Expand in powers of We obtain Lie algebra invariants (Casimirs) The degree of Casimir depends on g
Example: and . This can be used to prove that two partition functions are different Same PFs at but not 5 [Grushevsky, Salvati Manni ’08] Consider
More examples: c=24
Systematic procedure Different factorizations different Casimirs In particular, all independent Casimirs for adjoint representation can be obtained Lie algebra machinery… The affine symmetry is uniquely determined by the partition functions
Distinguishing reps? Example: overall spin flip in cannot be detected by PFs We cannot generate the whole algebra of Casimir invariants from PFs Evidence that representation content can be distinguished by PFs (up to Lie algebra outer automorphisms)
PARTITION FUNCTIONS AND MODULAR FORMS
PFs and modular forms Riemann surface of genus Riemann period matrix Genus partition function for MCFT Modular form of weight c/2 Example:
PFs and modular forms Genus PF for MCFT (centr. charge ) Modular properties Factorization properties
PFs and modular forms must satisfy some basic constraints (factorisation, modular properties) Finite number of parameters determine . Ex.: for : no free parameters : 1 parameter (number of currents )
PFs and modular forms Consequence: all invariants from depend on these parameters Example: with currents
Conclusions and to do From partition functions we can reconstruct the affine symmetry of a CFT Do PF’s determine representations? Partition functions of meromorphic CFT depend on finite number of parameters New consistency conditions on PFs?
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